Hindawi Publishing Corporation

Advances in Civil Engineering

Volume 2012,Article ID391214,10 pages

doi:10.1155/2012/391214

Research Article

APrediction Method of Tensile Young’s Modulus of

Concrete at Early Age

Isamu Yoshitake,

1

Farshad Rajabipour,

2

Yoichi Mimura,

3

and AndrewScanlon

2

1

Department of Civil and Environmental Engineering,Yamaguchi University,Ube,Yamaguchi 755-8611,Japan

2

Department of Civil and Environmental Engineering,Pennsylvania State University,University Park,PA 16802,USA

3

Department of Civil and Environmental Engineering,Kure National College of Technology,Kure,Hiroshima 737-8506,Japan

Correspondence should be addressed to Isamu Yoshitake,yositake@yamaguchi-u.ac.jp

Received 7 March 2011;Accepted 22 September 2011

Academic Editor:Kent A.Harries

Copyright © 2012 Isamu Yoshitake et al.This is an open access article distributed under the Creative Commons Attribution

License,which permits unrestricted use,distribution,and reproduction in any medium,provided the original work is properly

cited.

Knowledge of the tensile Young’s modulus of concrete at early ages is important for estimating the risk of cracking due to restrained

shrinkage and thermal contraction.However,most often,the tensile modulus is considered equal to the compressive modulus and

is estimated empirically based on the measurements of compressive strength.To evaluate the validity of this approach,the tensile

Young’s moduli of 6 concrete and mortar mixtures are measured using a direct tension test.The results show that the tensile

moduli are approximately 1.0–1.3-times larger than the compressive moduli within the material’s ﬁrst week of age.To enable

a direct estimation of the tensile modulus of concrete,a simple three-phase composite model is developed based on random

distributions of coarse aggregate,mortar,and air void phases.The model predictions show good agreement with experimental

measurements of tensile modulus at early age.

1.Introduction

An accurate estimation of the Young’s modulus is important

for proper structural design of concrete members,and ensur-

ing their serviceability,such as controlling deﬂections and

crack widths.In particular,the time-dependent development

of the tensile Young’s modulus at early ages is needed for

estimation of the tensile stresses that are generated due

to restrained thermal and hygral shrinkage.These tensile

stresses may lead to premature cracking of concrete mem-

bers.Currently,the tensile modulus is assumed to be equal

in value to the compressive modulus and is estimated using

empirical correlations based on the compressive strength

of concrete [1,2].The Architectural Institute of Japan

(AIJ) [3] points out that employing the tensile modulus

is more appropriate for estimation of the risk of early-

age cracking;however,the speciﬁcation indicates that the

compressive modulus may be used instead of the tensile

modulus because investigations dealing with the tensile

modulus are currently insuﬃcient.Since the tensile behavior

of concrete is more signiﬁcantly aﬀected by the presence of

ﬂaws (e.g.,microcracks or large capillary pores common in

early-age concrete),it is important to develop tools to predict

or measure the tensile properties more accurately.

Direct tension tests have been conducted in earlier

studies to investigate the tensile strengthand the tensile strain

capacity of concrete.Although the tensile moduli can be

obtained fromthe linear portion of the stress-strain diagram

in these reports,the focus of these earlier studies has been

primarily on mature concrete.As such,little information

is available on the early-age (i.e.,less than 28 days) tensile

modulus and its development with time.In addition,a

reliable model to aid design engineers in estimating the

tensile modulus based on concrete’s proportions and age

does not currently exist.

Xie and Liu [4] conducted a direct tension test using

small and large specimens of mature concrete with var-

ious aggregate sizes,and measured the tensile strength,

strain capacity,and Young’s modulus.They observed that

increasing the maximum aggregate size does not have a

proportional impact on tensile strength and tensile modulus

of concrete.Oluokun et al.[5] researched the compressive

2 Advances in Civil Engineering

Young’s modulus and Poisson’s ratio of early-age concrete.

They concluded that the compressive modulus is propor-

tional to the 0.5 power of the compressive strength,and

the ACI 318 formula for estimation of the compressive

modulus is valid after the age of 12 hours.Hagihara et al.

[6] investigated the tensile creep of high-strength concrete

at early age,and reported that the tensile Young’s moduli

are approximately 15%higher than the compressive moduli.

This is an important conclusion and should be evaluated

for other concretes with normal strength.Swaddiwudhipong

et al.[7] investigated the mechanical properties of con-

crete containing ground granulated blast furnace slag and

pulverized fuel ash.They reported that the tensile moduli

of all concretes tested had correlated well with the tensile

strength;although no predictive formula is presented for

the tensile modulus.Aoki et al.[2] also conducted direct

tension test in order to obtain tensile strength and Young’s

modulus of mature concrete and found that the tensile

modulus is 9–12% higher than the compressive modulus

for these concretes.Bissonnette et al.[8] researched the

tensile creep of concrete at early age,and presented some

measurements of the tensile Young’s modulus at the age of

7 and 28 days.In an earlier work [9],the authors investigated

the tensile Young’s moduli by using a direct tension test,

and presented a composite model derived from the Hirsh

model [10] to predict the tensile modulus.The model

showed good agreement with the experimental data as well as

other composite models oﬀered by Counto [11] and Hashin

[12].Recently,Mihashi and Leite [13] presented a state-

of-the-art report on early age cracking of concrete and its

mitigation techniques that includes some information on the

mechanical properties of concrete at early ages.

As described above,the investigations focusing on the

tensile Young’s modulus of early-age concrete are few,and

more experimental data will be needed to establish reliable

predictive correlations for estimation of the tensile modulus.

The present paper reports laboratory measurements of the

tensile modulus within the ﬁrst 7 days of hydration.Three

concrete mixtures (with diﬀerent W/C) were tested.In

addition,to evaluate the eﬀect of aggregate size,duplicate

concrete mixtures were prepared and sieved before setting

using a 5 mm mesh sieve.The resulting mortars were

tested to determine their tensile Young’s modulus.Using the

measurements results,a composite model was developed and

calibrated which can serve as a simple method for estimating

the tensile Young’s modulus of concrete.

2.Experimental Program

2.1.Materials and Mix Proportions of Concrete.This study

employed ordinary Portland cement with a density of

3.14 g/cm

3

.Tables 1 and 2 provide the details of the cement

and aggregate used.Proportions of the six concrete and

mortar mixtures tested in this study are given in Table 3.

The proportions of the concrete were designed by referring

to mixture proportions used in a ready mixed concrete

plant in Japan.Mortars were obtained by sieving plastic

concrete mixtures as discussed above.This was done to

duplicate the mechanical properties (i.e.,tensile modulus)

Table 1:Physical and chemical compositions of cement.

Ordinary Portland cement

Density 3.14 g/cm

3

Blaine ﬁneness 3340 cm

2

/g

Setting time start-end 2 h 26 m–3 h 34 m

Comp.strength at 3 days 30.8 MPa

at 7 days 46.3 MPa

at 28 days 63.6 MPa

Chemical compositions

CaO 64.5%

SiO

2

20.5%

Al

2

O

3

5.7%

Fe

2

O

3

2.9%

MgO 1.27%

SO

3

2.15%

Cl

−

0.009%

Loss of ignition 1.89%

Table 2:Properties of aggregate.

Fine agg.S Crushed rock G

Materials Sea sand Andesite

Density 2.56 g/cm

3

2.73 g/cm

3

Fineness modulus 3.36 6.66

Absorption 1.3% 1.3%

Size (max.– min.) 5 mm 20–5 mm

of the mortar portion of the concretes as closely as possible.

These results are needed to develop the composite model

as discussed in Section 3.In all mixtures,proper dosages of

air entraining and water reducing admixtures were used to

ensure consistency and workability of the concrete (Table 3).

2.2.Test Methods and Specimens.Figure 1 shows the direct

tension apparatus used in this study.Figure 2 shows the

geometry of the dog-bone specimens tested.To reduce

bending moment during test,the ends of the dog-bone

specimens were not fully ﬁxed but were allowed rotational

freedom.The direct tension apparatus manually provides

a tensile force to a specimen using a lever.Overall,an

approximately constant strain rate of 2 to 3

×

10

−

6

/sec

was applied.To measure tensile strain of specimens,an

embedded strain sensor was used which includes an electrical

resistance wire strain gage of 60 mm long coated by epoxy

resin with tensile modulus of 2.8 kPa as shown in Figure 2.

The overall sensor size was 120

×

10

×

3mm;as such the

area ratio of the sensor to concrete was smaller than 1.4%

to ensure that the sensor has little inﬂuence on the behavior

of specimens when subjected to tensile force.While this

setup was used to measure the tensile modulus,it may not

be suitable for measurement of the tensile strength since a

number of breaks occurred within the end zones of the dog-

bone specimens (Figure 3).

Advances in Civil Engineering 3

Table 3:Mixture proportions of concrete.

ID W/C (%) Water (kg/m

3

) Cement (kg/m

3

) S (kg/m

3

) G (kg/m

3

) WRA (kg/m

3

) Air (%)

O57 57 165 290 812 1030 2.9 4.5

O57m

#1

57 265 466 1304 — 4.7 —

#2

O39 39 169 434 790 933 4.3 3.8

O39m

#1

39 257 659 1200 — 6.5 —

#2

O25 25 170 680 694 818 6.8 3.6

O25m

#1

25 243 971 991 — 9.7 —

#2

#1

Mortar is made fromwet screening of concrete (maximummesh size:5 mm).

#2

Air content of mortar was not measured because it is diﬃcult to obtain wet-screened mortar volume required for the test.

Chain Chain

Load cell

Load

Pin

Specimen

Grip

Lever

Figure 1:Direct tension test using a dog-bone-shaped specimen.

10

15 70 120 70 15

75 100

20 80 20

20

t

=

3

Epoxy resin Strain gage (60 mmlong)

(unit:mm,1 mm

=

0.0394 in.)

Sensor

Specimen

Sensor

Figure 2:A dog-bone-shaped specimen and an embedded sensor.

3.Prediction of the Young’s Modulus Using

a Composite Model

In addition to the tensile modulus,the compression and

indirect tension tests were conducted using cylindrical spec-

imens.The compressive modulus of concrete was obtained

using an extensometer equipped with 2 displacement gages,

and the modulus of mortar was measured using 2 wire strain

gages 30 mm long.The cylindrical specimens (diameter

×

height) tested were 100

×

200mm for concrete,and 50

×

100mm for mortar.Three cylindrical specimens were used

for each test per each mixture,and the average of the three

measurements was used.The tests were performed at ages 1,

2,3,and 7 days.

3.1.Determination of the Tensile Young’s Modulus.To mea-

sure the tensile modulus,the tensile force is applied to the

specimen at strain intervals of 10

×

10

−

6

.In order to prevent

failure of specimen,the maximumstrain during the test is set

at 60

×

10

−

6

.Since the plastic strain of concrete at early age

Figure 3:Typical failure within the end zone of the dog-bone-

shaped specimen.

Young’s modulus

Tensilestress(MPa)

×

10

−

6

1010 20 30

Strain at loading

Residual strain

Loading

Removing load

Tensile strain

Figure 4:Evaluation method for tensile Young’s modulus.

may comprise a high percentage of the total strain measured,

the tensile modulus is obtained fromthe unloading branches

of the stress-strain relation as shown in Figure 4.The slope

of the unloading branches after the specimen was loaded

to 10

×

10

−

6

,20

×

10

−

6

,and so forth,is determined and

averaged to obtain the tensile modulus of the specimen.The

force is measured twice at each strain level using a load cell

4 Advances in Civil Engineering

(Tension)

(Tension)

(Compression)

(Compression)

E

1

V

1

E

2

V

2

E

i

V

i

E

n

V

n

(a)

E

1

V

1

E

2

V

2

E

i

V

i

E

n

V

n

(Tension)

(Compression)

(Compression)

(Tension)

(b)

(Tension)

E

m

V

m

E

g

V

g

(Tension)

(Compression)

(Compression)

(c)

E

m

V

m

E

g

V

g

(Tension)

(Tension)

(Compression)

(Compression)

(d)

Figure 5:Typical composite models for predicting Young’s modulus:(a) Parallel model,(b) Series model,(c) Counto model,and (d) Hashin

model.

with an accuracy of 0.1 kNand capacity of 200 kN.As will be

discussed later,the maximumresidual strain after unloading

of each specimen was measured as 3

×

10

−

6

.In each case,

concrete and mortar specimens were made from the same

batch using the sieving procedure mentioned above.

As mentioned earlier,the tensile Young’s modulus of

concrete is often assumed to be equal in value to the

compressive modulus.In addition,the compressive modulus

of concrete is frequently estimated based on empirical corre-

lations with concrete compressive strength.The compressive

modulus of concrete has also been related to the volume and

mechanical properties of concrete’s constituents (aggregates,

paste,etc.) using some classical composite models such

as those presented by Zhou et al.[14],Topc¸u [15],and

Yoshitake et al.[9].A brief overview of these models

is provided below.

The typical composite models for estimation of the

elastic modulus are illustrated in Figure 5.These include (a)

the Parallel model,(b) the Series model,(c) the Counto

model [11],and (d) the Hashin model [12];as represented

by the following:

Parallel model:E

=

n

i

=

1

E

i

∙

V

i

,

(1)

Series model:

1

E

=

n

i

=

1

V

i

E

i

,(2)

Advances in Civil Engineering 5

E

a

v

a

E

m

v

m

E

g

v

g

Parallel model

Seriesmodel

(Tension)

(Compression)

(Compression)

(Tension)

E:Young’s modulus of each material

v:volume of each element

a:air,m:mortar,g:coarse agg

E

m

v

m

Figure 6:An example of a simple composite model using parallel

and series models.

Counto model:

E

E

m

=

1 +

V

g

V

g

−

V

g

+E

m

/

E

g

−

E

m

,

(3)

Hashin model:

E

E

m

=

V

m

E

m

+

1 +V

g

E

g

1 +V

g

E

m

+V

m

E

g

,(4)

where E

i

and V

i

represent the Young’s modulus and the

volume fraction of concrete constituents (e.g.,mortar,coarse

aggregate,etc.),n is the number of constituents,and the

subscripts m and g refer to mortar and coarse aggregate,

respectively.The Counto and Hashin models are based on

a 2-phase composite (mortar and aggregate).While these

models are generally more accurate than simple parallel and

series models,they may estimate the Young’s modulus inap-

propriately for concretes containing high aggregate volumes

or high air content.

In the present work a new triphase model is proposed

based on random distribution of elements within a 2-

dimensional 80

×

80 grid (Figure 6).Each element in

the model is composed of mortar,coarse aggregate,or

air.The number elements corresponding to each phase is

proportional to the volume fractionof that phase in concrete.

Elements are placed randomly in the model using a Monte

Carlo procedure.To determine the tensile modulus of the

grid,simple micromechanical calculations are performed

based on the series and parallel models.First the tensile

modulus of each row of elements is determined using

the parallel model and then the modulus of the grid is

determined by combining all rows using the series model.

Alternatively,the modulus of each column can ﬁrst be

determined using the series model and then the columns are

combined using the parallel model.

4.Experimental Results and Discussion

4.1.Evaluation of the Reliability of the Embedded Strain

Sensor.It is important to ensure proper measurements of

the tensile strain using the embedded strain sensor.For this

purpose,a dog-bone concrete specimen is tested in tension

0

10

20

30

40

50

0 10 20 30 40 50

Surfacestrain

×

10

−

6

Surface strain

Inside strain

×

+

Inside strain

×

10

−

6

Figure 7:Comparison of inside strain and surface strain.

and the tensile strain is measured by both the embedded

gage as well as 2 wire strain gages mounted on the surface of

the concrete specimen.Figure 7 presents the results showing

that the internally measured strain is practically equal to the

surface measured strains.This implies that the tensile stress is

applied uniformly to the specimen and the embedded sensor

can be used to monitor concrete’s tensile strain.Based on this

conclusion,further measurements in this study employ only

the embedded sensor.

4.2.Tensile Stress-Strain Responses.Figure 8 presents exam-

ples of the tensile stress-strain responses of each concrete.As

shown,the slope of each stress-strain regression line (i.e.,the

tensile modulus) develops with increasing age and reducing

the water-cement ratio (W/C) of concrete.Figure 8(a) shows

the stress-strain response at 1 day;the results indicate that

the residual strain after unloading of specimens is zero (i.e.,

plastic strain at age of 1 day when specimen are loaded to

60

×

10

−

6

is negligible).Note that the response of mixture

O57 (W/C

=

57%) at 1 day could not be obtained because the

concrete was too weak to allowperforming the direct tension

test.Figure 8(b) presents the stress-strain responses at 7 days;

the maximumresidual strain after loading specimens to 60

×

10

−

6

is 3

×

10

−

6

corresponding to the mixture O39.For this

mix,had the loading branches of the stress-strain response

been used to determine the modulus,the tensile modulus

would be estimated as 31.2 GPa,comparing with 33.0 GPa

obtained from using the unloading stress-strain branches.

The ratio of the modulus obtained by the two methods is

approximately 0.95.

4.3.Time Dependent Development of the Mechanical Prop-

erties.The results of the compressive and splitting tensile

strength measurements of the three concrete mixtures are

presented in Figure 9.At 7 days,the concrete mixtures have

compressive strengths in the range of 20 to 45 MPa,and

splitting tensile strengths in the range of 2 to 3.3 MPa.The

time-dependent compressive and tensile Young’s moduli are

6 Advances in Civil Engineering

0

0.5

1

1.5

2

0 10 20 30 40 50 60 70

Tensilestress(MPa)

Strain

O39

O39

O25

O25

×

10

−

6

(a)

0 10 20 30 40 50 60 70

Tensilestress(MPa)

Strain

O57

0.5

1

1.5

2

0

O25

O39

O57

O39

O25

×

10

−

6

(b)

Figure 8:Tensile stress-strain responses:(a) age of 1 day and (b) age of 7 days.

0

10

20

30

40

50

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8

ft

(MPa)

Age (day)

O57,tensileO57,compressive

O39,tensileO39,compressive

O25,tensileO25,compressive

f

c

(MPa)

Figure 9:Strengths versus age of concrete.

presented in Figure 10.The compressive moduli in the graph

demonstrate the secant moduli under 33%of the maximum

stress.Based on Figure 10,the tensile Young’s moduli are

approximately 1.0–1.3-times larger than the compressive

moduli.The result may be caused by diﬀerent response

to stress applied to each specimen,such that the tensile

stresses were less than 10%of the compressive stresses.This

diﬀerence can be especially signiﬁcant at early ages when the

large macropores dictate the tensile response of concrete.

A comparison between the tensile modulus of concrete

and the corresponding mortar specimens are provided

in Figure 11.The results indicate that the concrete tensile

modulus is always higher than the mortar modulus due to

the stiﬀness provided by the coarse aggregates in concrete.

However,the diﬀerence is narrowed as concrete ages due to

hydration of cement which results in an increased stiﬀness of

the mortar.

15

20

25

30

35

40

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8

Et

(GPa)

Ec

(GPa)

Age (day)

O57,tensileO57,compressive

O39,tensileO39,compressive

O25,tensileO25,compressive

Figure 10:Young’s moduli versus age of concrete.

15

20

25

30

35

40

15

20

25

30

35

40

0

1 2 3 4 5 6 7 8

MortarEt

(GPa)

ConcreteEt

(GPa)

Age (day)

O57,mortar

O39,mortar

O25,mortar

O57,concrete

O39,concrete

O25,concrete

Figure 11:Tensile Young’s modulus predicted by employing grids

of diﬀerent ﬁneness.

Advances in Civil Engineering 7

23

24

25

26

27

28

29

10

×

10

20

×

20

30

×

30

40

×

40

50

×

50

60

×

60

70

×

70

80

×

80

90

×

90

100

×

100

PredictedEt

(GPa)

Mesh size

E

m

:19 GPa

E

g

:36 GPa

V

a

:4.5%

V

g

:37.7%

Average

10

∼

100

10

∼

100

Figure 12:Comparison of experiment and predicted tensile

Young’s modulus.

5.Prediction of the Tensile

Young’s Modulus at Early Age

5.1.Input Data for Development of the Simpliﬁed Composite

Model.The tensile Young’s moduli of coarse aggregates

and mortar are needed as input parameters for use in the

composite models.While the moduli of mortar is directly

measured (Figure 11),it is diﬃcult to directly obtain the

tensile modulus of the aggregate due the size and number

of specimens required for tensile testing,diﬃculty in using a

proper tensile grip method of the rock specimen,as well as

the variability in the material properties due to layering and

impurities of the rock.Thus,the tensile Young’s modulus of

the coarse aggregate is indirectly calculated by employing the

Counto and Hashin models in the present study.The study

estimates the tensile Young’s modulus (E

g

) of 36.0 GPa from

the experimental result of O25 at age of 7 days,for which

the modulus of mortar was almost equal to the modulus of

concrete.That is,the estimated value to the modulus is little

aﬀected by the models used for obtaining the modulus of the

aggregate.Similar values are obtained based on testing O25

at earlier ages or by testing the other two mixtures.

Table 4 provides the input data for the composite model.

The volume fractions of each component (mortar,coarse

aggregates,and air) are obtained from the mix proportions

given in Table 3.To determine the appropriate number

of elements in the model,the study estimates the tensile

Young’s moduli of concrete by employing models of various

mesh sizes.Figure 12 presents the resulting tensile moduli

predicted as a function of mesh ﬁneness.The graph shows

the maximum,minimum,and average moduli obtained

from 10 consecutive simulations in each model.Based on

these results,a 80

×

80 model was chosen for the remaining

simulations in this work.

5.2.Quality of the Model Predictions.Figure 13 shows the

tensile Young’s moduli of all concrete specimens predicted

by using the proposed composite model.Each bar presents

the average of 10 simulations for each concrete mixture

and at each age.The bar graph indicates that the predicted

0.6

0.7

0.8

0.9

1

1.1

15

20

25

30

35

40

O57(2)

O57(3)

O57(7)

O57(28)

O39(1)

O39(2)

O39(3)

O39(7)

O25(1)

O25(2)

O25(3)

O25(7)

Et

(GPa)

Ratio

Ratio (experiment/predicted)

Experiment

Predicted

Figure 13:Tensile Young’s modulus predicted by employing grids

of diﬀerent ﬁneness.

15

20

25

30

35

40

1.5 2.5 3.5 4.5

Et

ofmortar(GPa)

E

t

(7)

=

4.3C/W+18.1

E

t

(1)

=

4.3C/W+7.9

E

t

(2)

=

4.3C/W+11.9

C/W

E

t

(3)

=

4.3C/W+15.2

Figure 14:Relations between cement-water ratio and tensile

Young’s moduli of mortar.

Young’s moduli are in good agreement with the experimental

values;the ratio of the two is in range of 0.90 to 1.07.This

implies that the tensile modulus of concrete can be predicted

appropriately by employing the composite model when the

volume fraction of constituents is known.

5.3.Empirical Formula for the Tensile Modulus of Mortars

in This Study.Considering that the Young’s modulus of

coarse aggregate and air content of concrete are age-

independent,the Young’s modulus of concrete at early age

may be predicted if the modulus of mortar can be estimated

appropriately.For the mortar studied in this work,empirical

correlations between the experimental measurement mod-

ulus,the W/C,and the age of mortars are established as

presented in Figure 14.The ﬁgure shows a linear correlation

between the inverse water to cement ratio (C/W) and the

tensile moduli of mortar (E

t

) at ages of 1,2,3,and 7 days.

Interestingly,the slope k

i

of all regression lines in the graph

is approximately 4.3 GPa:

Relation of E

t

−

C

W

:E

t

=

k

i

∙

C

W

+C

i

,(5)

where E

t

(GPa) is the tensile modulus of mortar,and C

i

(GPa) is an age-dependent parameter in each regression line.

8 Advances in Civil Engineering

Table 4:Input data for the composite model.

Mix.ID V

m

V

g

V

a

E

t

of mortar (GPa) shown in Figure 11

E

g

(GPa)

1 day 2 days 3 days 7 days

O57 57.7% 37.7% 4.5% N/A 19.0 22.4 25.6 36.0

O39 62.0% 34.2% 3.8% 20.0 24.2 27.0 30.0 36.0

O25 66.4% 30.0% 3.6% 23.7 29.1 32.3 35.3 36.0

7.9

11.9

15.2

18.1

6

8

10

12

14

16

18

20

22

0 1 2 3 4 5 6 7 8 9 10

CoeﬃcientCi

(GPa)

Age T (day)

E

t

(T)

=

4.3C/W+C

i

R

=

0.93

C

i

(T)

=

21T

T

+1.5

Figure 15:Coeﬃcient C

i

of the regression line versus age of mortar.

The change in C

i

with age of mortar is shown in Figure 15.A

Goral curve [16],which is often used for estimating concrete

strength development with age,is ﬁt to the data points

resulting in the following:

Coeﬃcient C

i

with age:C

i

=

21T

T +1.5

,(6)

where T is mortar age in days.Combining (5) and (6) results

in

Tensile Young’s modulus of mortar:

E

t

(

T

)

=

4.3

C

W

+

21T

T +1.5

,

(7)

where E

t

is the estimated tensile Young’s modulus of mortar

(GPa).

It must be noted that similar to modulus of concrete,

the tensile modulus of mortar is a function of the volume

fraction and stiﬀness of sand,volume fraction and modulus

of cement paste (itself a function of age and W/C),and

the air content of the mortar.By accounting for age and

W/C,(7) can provide an estimate for the tensile modulus

of mortars with similar volume fraction and stiﬀness of

the ﬁne aggregates.Young’s modulus of coarse aggregate

can be considered as an inﬂuencing factor to the tensile

modulus of concrete,so this equation including the eﬀect of

ﬁne aggregate may be useful for normal concrete using sea

sand when an appropriate value of the modulus for coarse

aggregate is provided.To be applicable to mortars other than

those used here,the most signiﬁcant remaining parameter

is the volume fraction of sand which must be taken into

consideration.

5.4.Prediction of the Tensile Modulus of Concrete by the

Composite Model.By combining (7) with the composite

15

20

25

30

35

40

15 20 25 30 35 40

Prediction(GPa)

Experiment (GPa)

Present study

Aoki

Xie and Liu

Swadddiwdhipong

+15%

−

15%

Figure 16:Comparison of experiments and predictions.

model (Figure 6),the age-dependent tensile moduli of

concrete can be estimated from its mixture proportions

(i.e.,volume fraction of constituents) and the aggregate

modulus.Figure 16 presents a comparison between the

model predictions and the experimental data fromthis study

as well as those fromprevious investigations [2,4,7].Herein,

these predicted data were obtained with the assumption that

the aggregate properties in other studies are equal to the

values employed in this study,because the modulus of coarse

aggregate used is not reported in the previous investigations.

The ﬁgure shows that the proposed method can predict the

tensile modulus of concrete,with reasonable accuracy,solely

based on the mixture proportions.

Figure 17(a) describes characteristics of tensile moduli of

concrete with W/C

=

55%and coarse aggregates of diﬀerent

volume fractions.The vertical axis in the graph presents the

tensile Young’s moduli ratio of concrete to coarse aggregate.

Herein,the model results are obtained fromthe assumptions

shown in Table 5.The ﬁgure demonstrates that the tensile

modulus of concrete increases with increasing the volume of

coarse aggregates since the aggregates have a higher Young’s

modulus than the mortars.It is also noted that tensile moduli

of concrete having more coarse aggregates (i.e.,larger vol.

fractions) than the solid volume content of coarse aggregates

[17] are unavailable.Figure 17(b) presents a similar set of

curves corresponding to concrete with W/C

=

30%.Both

ﬁgures describe that the tensile modulus of concrete develops

rapidly at early age and gradually plateaus at later ages.

Advances in Civil Engineering 9

Table 5:Conditions for simulation of tensile Young’s modulus.

Composite model See Figure 6

Elements 80

×

80

Air content 4.5%

Tensile Young’s modulus of mortar see (7)

Tensile Young’s modulus of coarse aggregate 36 GPa

Solid volume content of coarse aggregate 60%

Tensile Young’s modulus of concrete

Average of 10

simulations

40

50

60

70

80

90

100

0 20 40 60 80 100

28 d

3 d

1 d

2 d

5 d

7 d

14 d

91 d

Solid volume percentage

W/C:55%

V

a

:4.5%

RatioofEt

(conc./coarseagg.)

(%)

Volume of coarse aggregate

(%)

(a)

40

50

60

70

80

90

100

0 20 40 60 80 100

28 d

3 d

1 d

2 d

5 d

7 d

14 d

91 d

Solid volume percentage

W/C:30%

V

a

:4.5%

RatioofEt

(conc./coarseagg.)

(%)

Volume of coarse aggregate

(%)

(b)

Figure 17:Prediction results for tensile Young’s modulus of

concrete:(a) W/C

=

55%and (b) W/C

=

30%.

6.Conclusions

This paper describes the experimental measurement of

tensile Young’s modulus of concrete at early age using a direct

tension setup.Moreover,a predictive composite model was

developed to estimate the age-dependent tensile modulus

of concrete using the volume fractions and properties of

the constituents.The main conclusions are summarized as

follows.

(1) The tensile stress-strain response of concrete was

observed to be very linear even at early ages (e.g.,1

day old).The residual strains after repeated loading

of specimens up to strains of 60

×

10

−

6

were neg-

ligible.The tensile Young’s modulus obtained from

the stress-strain response develops according to the

water-cement ratio and the age of concrete.

(2) The tensile modulus of concrete is approximately

1.0–1.3-times larger than its compressive modulus

within the material’s ﬁrst week of age.As such,

estimation of the tensile modulus based on empirical

correlations with the compressive strength of con-

crete can be inaccurate.

(3) The age-dependent tensile moduli of concrete could

be predicted appropriately by the proposed com-

posite model as long as the volume fractions of

coarse aggregates,mortar,and air,and the modulus

of aggregates are known.In comparison with the

experimental results,the model predictions showed

accuracy better than

±

15%.

References

[1] A.M.Neville,Properties of Concrete:Fourth and Final Edition,

Pearson/Prentice Hall,1995.

[2] Y.Aoki,K.Shimano,D.Iijima,and Y.Hirano,“Examination

on simple uniaxial tensile test of concrete,” Proceedings of the

Japan Concrete Institute,vol.29,no.1,pp.531–536,2007

(Japanese).

[3] Architectural Institute of Japan,Recommendations of Practice

of Crack Control in Reinforced Concrete Buildings-Design and

Construction,Architectural Institute of Japan,2006.

[4] N.Xie and W.Liu,“Determining tensile properties of mass

concrete by direct tensile test,” ACI Materials Journal,vol.86,

no.3,pp.214–219,1989.

[5] F.A.Oluokun,E.G.Burdette,and J.H.Deatherage,“Elastic

modulus,Poisson’s ratio,and compressive strength relation-

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3–10,1991.

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[10] T.J.Hirsh,“Modulus of elasticity of concrete aﬀected by elastic

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[11] U.J.Counto,“The eﬀect of the elastic modulus of the

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1964.

[12] Z.Hashin,“The elastic moduli of heterogeneous materials,”

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150,1962.

[13] H.Mihashi and J.P.D.B.Leite,“State-of-the-art report on

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10 Advances in Civil Engineering

[14] F.P.Zhou,F.D.Lydon,and B.I.G.Barr,“Eﬀect of coarse

aggregate on elastic modulus and compressive strength of high

performance concrete,” Cement and Concrete Research,vol.25,

no.1,pp.177–186,1995.

[15] I.B.Topc¸u,“Alternative estimationof the modulus of elasticity

for damconcrete,” Cement and Concrete Research,vol.35,no.

11,pp.2199–2202,2005.

[16] L.M.Goral,“Empirical time-strength relations of concrete,”

ACI Journal,vol.53,pp.215–224,1956.

[17] JIS A 1104,“Methods of Test for Bulk Density of Aggregates

and Solid Content in Aggregates,” Japan Industrial Standards

Committee,2006.

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